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A coding of hierarchical structure in finite directed tree graphs was introduced by Robert Horton in 1945; after a modification by Strahler in 1952, this has become a standard river network code defined as follows. Edges along a path adjoining a source (valence 1 vertex) to a junction (valence 3 or higher vertex) are coded as order 1. Such a path of order 1 edges is also referred to as an order 1 stream. Having defined edges and streams of order i, one now recursively defines edges and streams of order i+1 by the rule that the `order 1' streams of the tree obtained by pruning the streams of order i (and lower) are assigned order i+1. This code has helped to identify a number of naturally occurring patterns in river network structure as well as in other naturally occurring dendritic structures. Let Ti,j(s) denote the number of order j < i junctions in a stream s of order i and let denote the sample average over all such streams s of order i. Empirically it has been observed that is, allowing small-sample fluctuations, approximately a function of i-j for large classes of river networks. A calculation of Ronald Shreve published in 1969 revealed that ETi,j = ½2i-j for the critical binary Galton-Watson distribution. This is our starting point. In particular, we introduce a more general notion of stochastic self-similarity and show that within a class of Galton-Watson trees both this and the consequent mean Toeplitz property are characteristic of the critical binary offspring distribution. In addition, we obtain interesting conditioned limit theorems corresponding to the invariance of the critical binary branching process under a pruning dynamic in the space of finite rooted trees.
It is well known that fractional Brownian motion can be obtained as the limit of a superposition of renewal reward processes with inter-renewal times that have infinite variance (heavy tails with exponent α) and with rewards that have finite variance. We show here that if the rewards also have infinite variance (heavy tails with exponent β) then the limit Zβ is a β-stable self-similar process. If β≤α, then Zβ is the Lévy stable motion with independent increments; but if β> α, then Zβ is a stable process with dependent increments and self-similarity parameter H = (β- α+ 1)/β.
We prove almost sure limit theorems for weighted sums of i.i.d. random variables with weights having a very short span, so that rather strong moment conditions are needed. This complements results due to Li et al. and the first author.
We study the positive random measure , where denotes the family of local times of the one-dimensional Brownian motion B. We prove that the measure-valued process is a Markov process. We give two examples of functions for which the process is a Markov process.
In this paper we discuss stochastic differential delay equations with Markovian switching. These can be regarded as the result of several stochastic differential delay equations switching among each other according to the movement of a Markov chain. One of the main aims of this paper is to investigate the exponential stability of the equations.
We study the asymptotic behaviour of a system of interacting particles with space-time random birth. We have propagation of chaos and obtain the convergence of the empirical measures, when the size of the system tends to infinity. Then we show the convergence of the fluctuations, considered as cadlag processes with values in a weighted Sobolev space, to an Ornstein-Uhlenbeck process, the solution of a generalized Langevin equation. The tightness is proved by using a Hilbertian approach. The uniqueness of the limit is obtained by considering it as the solution of an evolution equation in a greater Banach space. The main difficulties are due to the unboundedness of the operators appearing in the semimartingale decomposition.
Let be a compact integral operator with a symmetric kernel h. Let , be independent S-valued random variables with common probability law P. Consider the n×n matrix with entries (this is the matrix of an empirical version of the operator H with P replaced by the empirical measure Pn), and let Hn denote the modification of obtained by deleting its diagonal. It is proved that the distance between the ordered spectrum of Hn and the ordered spectrum of H tends to zero a.s. if and only if H is Hilbert-Schmidt. Rates of convergence and distributional limit theorems for the difference between the ordered spectra of the operators Hn (or ) and H are also obtained under somewhat stronger conditions. These results apply in particular to the kernels of certain functions of partial differential operators L (heat kernels, Green functions).
A `skewing' method is shown to effectively reduce the order of bias of locally parametric estimators, and at the same time retain positivity properties. The technique involves first calculating the usual locally parametric approximation in the neighbourhood of a point x' that is a short distance from the place x where the we wish to estimate the density, and then evaluating this approximation at x. By way of comparison, the usual locally parametric approach takes x'=x. In our construction, x'-x depends in a very simple way on the bandwidth and the kernel, and not at all on the unknown density. Using skewing in this simple form reduces the order of bias from the square to the cube of bandwidth; and taking the average of two estimators computed in this way further reduces bias, to the fourth power of bandwidth. On the other hand, variance increases only by at most a moderate constant factor.
At extreme levels, it is known that for a particular choice of marginal distribution, transitions of a Markov chain behave like a random walk. For a broad class of Markov chains, we give a characterization for the step length density of the limiting random walk, which leads to an interesting sufficiency property. This representation also leads us to propose a new technique for kernel density estimation for this class of models.